Problem: Simplify and expand the following expression: $ \dfrac{3}{p - 10}+ \dfrac{1}{3p - 21}- \dfrac{3}{p^2 - 17p + 70} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $3$ out of denominator in the second term: $ \dfrac{1}{3p - 21} = \dfrac{1}{3(p - 7)}$ We can factor the quadratic in the third term: $ \dfrac{3}{p^2 - 17p + 70} = \dfrac{3}{(p - 10)(p - 7)}$ Now we have: $ \dfrac{3}{p - 10}+ \dfrac{1}{3(p - 7)}- \dfrac{3}{(p - 10)(p - 7)} $ The least common multiple of the denominators is: $ (p - 10)(p - 7)$ In order to get the first term over $(p - 10)(p - 7)$ , multiply by $\dfrac{3(p - 7)}{3(p - 7)}$ $ \dfrac{3}{p - 10} \times \dfrac{3(p - 7)}{3(p - 7)} = \dfrac{9(p - 7)}{(p - 10)(p - 7)} $ In order to get the second term over $(p - 10)(p - 7)$ , multiply by $\dfrac{p - 10}{p - 10}$ $ \dfrac{1}{3(p - 7)} \times \dfrac{p - 10}{p - 10} = \dfrac{p - 10}{(p - 10)(p - 7)} $ In order to get the third term over $(p - 10)(p - 7)$ , multiply by $\dfrac{3}{3}$ $ \dfrac{3}{(p - 10)(p - 7)} \times \dfrac{3}{3} = \dfrac{9}{(p - 10)(p - 7)} $ Now we have: $ \dfrac{9(p - 7)}{(p - 10)(p - 7)} + \dfrac{p - 10}{(p - 10)(p - 7)} - \dfrac{9}{(p - 10)(p - 7)} $ $ = \dfrac{ 9(p - 7) + p - 10 - 9} {(p - 10)(p - 7)} $ Expand: $ = \dfrac{9p - 63 + p - 10 - 9}{3p^2 - 51p + 210} $ $ = \dfrac{10p - 82}{3p^2 - 51p + 210}$